A-Level Unit Test: Trigonometry Small Angle Approximations - Jethwa Maths

A-Level Unit Test: Trigonometry

Small Angle Approximations

1. Given that is small and is measured in radians, use the small angle approximations to find an approximate

value of 1-4

2 3

(3)

2.

The

diagram

shows

triangle

ABC

in

which

angle

A

=

radians,

angle

B

=

3 4

radians

and

AB

=

1

unit.

a.

Use

the

sine

rule

to

show

that

AC

=

cos

1 - sin

(3)

b. Given that is a small angle, use the result in part (i) to show that, AC 1 + p + q2, where p and q are

constants to be determined.

(3)

3.

When

is

small,

show

that

the

equation

1+sin +tan 2 2 cos 3-1

can

be

written

as

1 1-3

(4)

b.

Hence

write

down

the

value

of

1+sin +tan 2 2 cos 3-1

when

is

small.

(1)

4a. When x is small, show that tan(3x) cos(2x) can be approximated by 3x ? 6x3

(3)

b. Hence, approximate the value of tan (0.3)cos(0.2)

(2)

c. Calculate the percentage error in your approximation.

(1)

Total marks: 20

Mark Scheme

1.

4

=

1-

(4)2 2

M1

3 = 3

M1

1-4 2 3

1-[1- (42)2] (2)(3)

82 62

4 3

M1

2a.

34

=

sin

1 (-34-)

M1

AC =

34 14 cos - 14 sin

M1

3 4

=

1 4

=

1 4

AC = 1

M1

cos-sin

2b.

AC

=

(1

+

(-

-

1 2

2))-1

M1

AC

=

1

+

(-1)(-

-

?

2)

+

(-1)(-2)(-

2

-

?

2)

+

....

M1

Therefore,

AC

1

+

+

3 2

2

M1

3a.

2cos

3

2(1

-

92)

2

=

2

-

92

M1

2cos 3 - 1 1 - 92 = (1 - 3)(1 + 3)

M1

1 + sin + tan 2 = 1 + + 2 = 1 + 3

M1

1+sin +tan 2 =

1+3

=1

2 cos 3-1 (1-3)(1+3) 1-3

M1

3b.

When

is

small,

1 1-3

1

M1

4a.

tan(3x)cos(2x) = 3x(1 - (2)2)

M1

2

= 3x(1 ? 2x2)

M1

= 3x ? 6x3

M1

4b.

x = 0.1, 3(0.1) ? 6(0.1)3 = 0.294

M1

4c.

tan(0.3)cos(0.2) = 0.3031701196

M1

%

error

=

0.3031701196-0.294 0.3031701196

?

100

M1

= 3.02% (to 2 decimal places)

M1

A-Level Unit Test: Trigonometry

Proof

1. Show that the equation tan 2x = 5 sin 2x can be written in the form (1 ? 5cos 2x)sin 2x = 0

(3)

2. Prove that tan x + cot x 2cosec 2x

(4)

3.

Prove

1-cos 2 sin 2

=

tan

(3)

4.

Prove

cosec

x

+

tan

x

=

cot

2

(4)

5. Show that cosec 2x + cot 2x = cot x

(4)

6. By writing 3x = (2x + x), show that 3x = 3sin x ? 4sin3 x

(4)

7. Use the identity cos2x + sin2x = 1 to prove that tan2x = sec2x ? 1

(2)

8. Show that (sin x + tan x)(cos x + cot x) (1 + sin x)(1 + cos x)

(3)

9. Show that sec2x ? sin2x tan2x + cos2x

(2)

10.

Prove

that

1-sin 2 -2 cos

sin

x

(3)

11. Prove that sin x + sin 2x + sin 3x sin2x (2cos x + 1)

(2)

12. Prove the identity 1-cos =

1+cos

2

2

(2)

Total marks: 36

Mark Scheme

1.

tan 2x = 5sin 2x

M1

sin 2 cos 2

=

5

sin

2

sin 2x = 5sin 2x cos 2x

M1

sin 2x ? 5sin 2x cos 2x = 0

(sin 2x)(1 ? 5cos 2x) = 0 (1 ? 5 cos 2x) sin 2x = 0

M1

2.

tan x + cot x = sin + cos = 2+ 2

M1

cos sin

2+ 2 =

1

?2

M1

2

=

2

2 cos

sin

M1

=

2 2

M1

= 2 cosec 2x

3.

1-cos 2 = 1-(1-22)

M1

sin 2

2

=

1-1+22 2

= sin

M1

cos

= tan x

M1

4.

cosec x + tan x = 1 + cos

sin sin

M1

= 1+cos

sin

M1

=

1+222-1 222

=

222 222

M1

=

cos2 sin2

=

cot

2

M1

5.

cosec 2x + cot 2x = 1 + cos 2

= 1+cos 2

sin 2 sin 2

M1

sin 2

= 1+(22-1)

M1

2 cos

= 22

= 2cossin cos

M1

sin

= cot x

M1

6.

sin (3x) = sin (2x + x)

= sin 2x cos x + sin x cos 2x

M1

= 2sinx cos x (cos x) + sin x cos 2x

= 2sin x cos2x + sin x(1 ? 2sin2x)

M1

= 2 sin x(1 ? sin2x) + sin x(1 ? 2sin2 x)

M1

= 2 sin x ? 2sin3 x + sin x ? 2sin3 x = 3sin x ? 4sin3x

M1

7.

cos2x + sin2x = 1

2 2

1

M1

2 + 2 = 2

1 + tan2x = sec2x Therefore, tan2x = sec2x - 1

M1

8.

= sin x cos x + sin x cot x + tan x cos x + 1 = sin x cos x + cos x + sin x + 1

M1

= sin x(cos x + 1) + cos x + 1 = (cos x + 1)(sin x + 1)

M1

9.

(LHS) 1 + tan2x ? (1 ? cos2x)

M1

= tan2x + cos2x

M1

10.

(LHS) = sin (1-2)

sin ( -2)

M1

= sin (1-sin 2)

M1

1-2 sin cos

=

sin (1-2) 1-sin 2

M1

= sin x

11.

(LHS)

=

2

sin

+3 2

cos

-3 2

+

sin

2x

M1

= 2 sin 2x cos (-x) + sin 2x

= 2 sin 2x cos x + sin 2x

M1

= sin 2x(2 cos x + 1) (RHS)

12.

(LHS) =

=

222 222

1-(1-222) 1+(222-1)

= tan2

2

M1 M1

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